« The economics of curling | Main | No two are alike »
The tennis ball problem
Natasha often says we should open more cans of tennis balls. Last night we were playing with eight balls and she wanted to play with twelve. Of course once all eight have been plowed into the net, you have to go collect at least some of them.
How many tennis balls should you play with?
Let's say you had many, many balls and you could open the cans for free and never run out. Opening a new can every four points (four balls fit in a can) would lead to a massive clean-up and carry problem at the end. Furthermore how much help is it having more balls? Once they hit the net you still have to deal with getting another ball into play. In other words, the real trick is to manage your stock well (read: aim for good volleys), not to just to speed up the flow of balls into the court.
Just one ball is not efficient, because when it falls out of play it is probably far from you. The greater the number of balls, the more likely at least one will be close.
Many problems in life, including those of dating, the number of children you should have, and optimal inventory management, resemble the tennis ball problem.
I do not know how to solve the tennis ball problem, but I feel that twelve balls is too many.
Posted by Tyler Cowen on February 16, 2006 at 06:29 AM in Sports | Permalink
TrackBack
TrackBack URL for this entry:
http://www.typepad.com/t/trackback/3576/4271128
Listed below are links to weblogs that reference The tennis ball problem :
» Think like an economist from Cosmic Variance
I learn a lot from reading Brad DeLong or the economically-inclined folks at Crooked Timber, but I get special enjoyment from Marginal Revolution. (Im actually not being sarcastic for once, in case its unclear.) Much like training in p... [Read More]
Tracked on Feb 16, 2006 11:37:40 AM
Comments
Twelve is far too many. You'll end up with a bunch of balls strewn around the court, with the consequent risk of serious injury. Also, some balls will get a lot more play than others, leading to dramatic pressure variations.
I'd say four is the optimal number, but your mileage may vary.
Posted by: Andy at Feb 16, 2006 7:55:35 AM
Become sufficiently good at tennis so that you (or your wife) never hit the ball into the net or out of bounds, and you'll only need one ball.
But that's rather glib and doesn't solve the problem.
Posted by: Dave at Feb 16, 2006 7:56:32 AM
Retrieving tennis balls is also exercise, no? I'd say you want about as many tennis balls as you can load in both your pockets without setting any on the ground to trip over or being uncomfortable. Probably 5 then (4 in the pockets and one to hit). Maybe a couple more if you have especially large pockets.
Posted by: Slocum at Feb 16, 2006 8:13:52 AM
I play a lot of tennis, so I think I have something to say on this topic.
Tennis is a great sport from the perspective of getting a lot of excercise, except when it isn't. According to an expert,
"Many studies have shown that among the various fitness parameters, anaerobic alactic motor abilities such as agility, speed, explosiveness, strength and power are best correlated with on-court performance."
When you add these anaerobic requirements to the obvious aerobic/cardiovascular benefits of running yourself ragged for a good long time, you've got yourself one hell of a workout. (And that's saying nothing about the signifcant types of mental engagement including concentration, strategy, excercising determination, etc.)
So what's the problem? Why isn't tennis a more popular means for getting fit, attracting tons of hardcore athletes into its ranks? (Compare cardio-boxing. Then shudder.)
Because tennis as its currently played at the entry level just isn't that good excercise. Especially during the beginning stages, your fantastic workout is often ruined by running after the stupid balls which are either in the net or (worse) on another court somewhere. Of course, you don't run after the balls. You walk to them. In a leisurely manner.
How many balls does it take to solve this problem? (BTW, the optimal solution is to have ball-children, like at tournaments.) First, contra Tyler, hitting balls into the net is not the big problem, since balls in the net generally do not pose a tripping threat and can safely be ignored.
Also, contra Tyler, I think 12 is the optimal answer, at least when not playing a competitive set. Each player can pocket four balls, while four lucky balls will get to roam free wherever they land. For competitive tennis, the right answer is eight, since the non-serving player doesn't need balls.
Economics. If you use new balls every time, a true luxury but arguably worth it, this costs out to $10 per session of tennis.
Enjoy!
Posted by: dr awkward at Feb 16, 2006 8:28:54 AM
"I do not know how to solve the tennis ball problem..."
Simple: Hire a ball boy.
Posted by: KipEsquire at Feb 16, 2006 8:37:25 AM
dr awkward -
Very good analysis. I would suggest, however, that another relevant factor is the number of courts surrounded by the perimeter fence. If there is merely one court, the players can get by with fewer than 12 balls while still maintaining high exercise quality, simply because errant balls can't go too far. If there are two or more courts, balls can travel farther, necessitating longer retrieval times, hence the need for more balls.
Posted by: Peter at Feb 16, 2006 9:18:26 AM
Twelve is ridiculous, unless you are in training and someone has a hopper and is consistently firing shots at you. In a match, 3 is the correct answer. Four is acceptable.
Without ball boys (or girls), the tendency for some balls to get used more than others, thus wearing out quicker leaving you with inconsistent bounce, is the bigger problem. Balls significantly wear out in a single set.
Given your premise, open new balls every game, and throw the old ones out.
Any balls not in or very close to the net or in a corner should be picked up on each point.
Posted by: theCoach at Feb 16, 2006 9:49:41 AM
...and the more I think about it, this problem is really the great problem of economics - using a set of limited parameters and solving the wrong problem.
Posted by: theCoach at Feb 16, 2006 9:53:08 AM
You don't have infinitely large pockets. Four is I think ideal. That means the greatest number of balls in your pockets at one time is 3. This is the question I want the answer to: how come there are only two tennis shops in the WashDC area, and one is run by incompetents and the other is run by price gougers?
Posted by: anderson@yahoo.com at Feb 16, 2006 10:07:38 AM
"I do not know how to solve the tennis ball problem, but I feel that twelve balls is too many."
Do what I do. Hit the balls over the fence.
Posted by: Robert Schwartz at Feb 16, 2006 10:25:56 AM
Tennis coaches over here usually have a large net of balls, maybe 50, so they don't have to interrupt instruction frequently to collect balls. It's quite a workable strategy.
Posted by: A Tykhyy at Feb 16, 2006 10:52:16 AM
Hmmmm. Regarding "optimal inventory management" and availability, are you suggesting I don't need my 32 pairs of scissors? Did my husband put you up to this?
Posted by: Kathleen Fasanella at Feb 16, 2006 10:56:31 AM
Thanks for your support. Tyler, please stop on the way home to buy more balls.
Posted by: Natasha at Feb 16, 2006 11:28:22 AM
Generally, you're only suppose to use three balls at any given time in a match play situation. The serving player has two and the receiving player one.
Posted by: Rob at Feb 16, 2006 11:32:53 AM
Well, if we want to think of it economically, tennis balls usually (or at least used to) come in packs of three. So playing with four requires opening another canister, at which point you might as well play with six.
Posted by: jvance at Feb 16, 2006 11:57:55 AM
You wrote: "Let's say you had many, many balls and you could open the cans for free and never run out."
Then, based upon your assumption, you don't have to pick up any. Just walk away at the end of your game.
Posted by: Dave Barnes at Feb 16, 2006 12:18:01 PM
I'm not a tennis player, but I am a mother, so I would suggest that what you need is only one ball and a small toddler (18 months to 2 years old) to happily run after it saying "Ball! Ball!" The toddler will fetch your ball, and you will get a workout from running after the toddler, and everyone involved will have a good time.
Posted by: Wacky Hermit at Feb 16, 2006 12:32:05 PM
Tyler, the solution to the tennis ball problem is to do as your wife says, but to talk about it occasionally. By experimenting with the increase from eight to twelve with enthusiasm, you defuse any defensiveness on her part and make her more open to considering that twelve balls may be too many. That way, if eight balls turns out to be more optimal, as determined by your joint observations, she can agree to that fact without having to admit that she was wrong and you were right.
Of course, it's also possible that you and she have different preferences with respect to searching for balls during a match versus clean-up, in which case you should take up racquetball.
Posted by: Matt at Feb 16, 2006 12:32:28 PM
Play badminton instead of tennis. The ball never gets very far away and the quicker pace makes for a more enjoyable game.
What the implications are for dating, number of children and inventory managment is left as an exercise for the reader.
Posted by: Johan Richter at Feb 16, 2006 1:14:43 PM
One does need to consider negative externalities, particularly if the players are not skilled. If one is playing in close proximity to others, more balls leads to more hitting which leads to more interference with others. From an impartial perspective where one considers the interests of all involved, the optimal number of balls probably goes up as skill level goes up, although it no doubt peaks somewhere between 4 and 10.
Posted by: alan wertheimer at Feb 16, 2006 1:56:44 PM
I find the standard 3 balls per can to work well. You can start off with two in your pocket, while you serve with the third. Assuming you miss the first serve, that ball will end up in a corner or at the net, and you can pull out one of the other two. Use that for the second serve and to play out the point, and then you have one left in your pocket. Then, you just need to get one of the other two balls from the court, in case you need to serve twice. The third ball can be picked up when it's convenient to do so, with less chance of posing a hindrance than if you had a lot more balls on the court.
Posted by: Joe at Feb 16, 2006 2:18:34 PM
Given that a tennis player almost never stands within a foot of the net, you should redesign the court so that it slopes up in the foot closest to the net on both sides. As long as you don't raise the net when you do this, you'll be able to continue playing normally and most balls will be returned to you by gravity.
Posted by: Peter Hakrins at Feb 16, 2006 2:19:46 PM
Yikes. If you need 12 balls let's organize a match and play for money, Tyler. Three is optimal, four is workable. Anything more than that is annoying. Anything less leads to too many delays, particularly on let services.
Posted by: Lance Knobel at Feb 16, 2006 3:10:12 PM
It depends on what you are doing. If you are playing a match, three balls is optimal. If not, get a hopper full of balls and just keep hitting until the hopper is empty.
Posted by: Kevin Nowell at Feb 16, 2006 3:56:13 PM
Bowl.
Posted by: eddie at Feb 16, 2006 5:07:34 PM
